I tend to like constructivist maths because it makes sense from a computational point of view. A boolean function may return True or False or it may not return at all (a member of the bottom set in Type theory terms) if it gets stuck in an infinite loop.

So I'm going to look at things from a computational point of view, to see if anything shakes loose. What might ~(A and ~A) represent computationally. The one thing I can think of is if A isn't a pure function, it is an unknown monad of some variety (there is some hidden state). A race condition or memory corruption that means when A is evaluated at different points in time it comes to a different value.

Going by the MWI hint, it also seems to be able to represent questions that aren't meaningful to ask. Did a particle travel through that path or did it not travel that path? Well it travelled through all possible paths, which is neither a yes or a no. You may want a logic that can represent such meaningless questions, because you can't apriori know that a question is meaningless.